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Solution to Problem3


*****

Problem3:

The curve,

> ;

                                     2
                              y = a x  + b x + c

passes through the points (x1,y1), (x2,y2), (x3,y3). Show that the coefficients a,b, and c are a solution of the system of linear equations whose augmented matrix is,

> ;

                            [  2                 ]
                            [x1     x1    1    y1]
                            [                    ]
                            [  2                 ]
                            [x2     x2    1    y2]
                            [                    ]
                            [  2                 ]
                            [x3     x3    1    y3]


SOLUTION:


Each pt. (x,y) on the plane produces the equation of the curve. i.e.,

> Eq := (x,y) -> a*x^2+b*x+c = y;

                                         2
                      Eq := (x, y) -> a x  + b x + c = y

since the curve must pass through each of the given points we have 3 equations in the unknowns a,b and c given by,

> Eq(x1,y1); Eq(x2,y2); Eq(x3,y3);

                                 2
                             a x1  + b x1 + c = y1

                                 2
                             a x2  + b x2 + c = y2

                                 2
                             a x3  + b x3 + c = y3

hence, the augmented matrix is the one specified in the problem.


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Tue Feb 8 17:03:34 EST 2000